After losing the second set, world number five Kuznetsova took complete control in the third, hammering the Spanish number one into submission after one hour 42 minutes.
After wasting several break points to get back on serve in the second game of the final set, Medina Garrigues took a medical time-out to treat her lower back.
The Spaniard's resistance crumbled after she surrendered her serve again in the next game to go 3-0 down and Kuznetsova cruised to victory, finishing the match with an ace.
On Saturday, world number nine Nadia Petrova overcame a slow start to beat Medina Garrigues 6-3, 6-4 in the opening singles rubber and Kuznetsova crushed 66th-ranked Lourdes Domingues Lino 6-3, 6-2 in 66 minutes to put the Russians in command.
"I felt nervous yesterday because I had not played Fed Cup for two years and today I think I was just too excited to play," Kuznetsova told Russian television NTV.
"In the second set I started to feel dizzy. In the third set I just relaxed and played much better."
In the semi-finals on July 14 and 15, the 2004 and 2005 champions will play the winners of the United States v Belgium tie. The Americans are leading 2-0 after the first day in Delray Beach, Florida.